Counting is such a basic tool to us that it is easy to assume that children know
what they are doing as they count, or what we mean when we ask an apparently
simple question like "How many?"
When working with children who have severe learning difficulties we need to look
closely at these assumptions. All too often we ask them to count and are frustrated
when they produce the wrong answer.
We need to help children learn from the process of counting.
In this article we find that counting is a complex skill.
We look at how children learn to count, describe in detail the "Parts of Counting"
and the "Principles of counting".
The ideas discussed will help teachers who need to focus on closer detail than
the National Curriculum P.O.S. in order to provide suitable approaches to help children
with Severe Learning Difficulties progress.
Learning the skills described in this article will be appropriate for the Numeracy Hour
for many children with very special needs.
Learning to Make it Count
As easy as 1 2 3 -- like talking and thinking most people seem to assume that the
skill of "counting" just happens. Those of us who work with children who
have severe learning difficulties are very aware that this is not the case.
Generally teachers of young children are very conscious of the skills which children
need to learn before they read. Pre-reading activities are well understood and a wide
range of materials are commercially available to help develop constituent skills for
Most maths schemes on the other hand give only a fairly cursory acknowledgement to
the constituent skills of learning to count and they move very quickly to using counting
as a tool for mathematical and arithmetic processes. Though there is research information
available that looks at how children learn to count there is less practical guidance for
teachers that illustrates how children learn the skill, or provides material to promote
the constituent parts of "counting."
Likewise the national curriculum assumes that children have usually made considerable
headway in mastering the skills of counting before they even start school. The programmes
of study do acknowledge some early aspects of learning, but to attain level 1 pupils need
to have mastered them and moved on to the use of numbers as tools.
The QCA target setting document "Supporting the Target Setting Process" was
published in 1998 and later P Levels described in QCA Guidance for teaching pupils
with learning difficulties (published in 2001) gave more guidance for early mathematical
learning than the national curriculum. P levels for Number, describe some
aspects of learning to count, as do differentiated criteria for levels 1 and 2 of the
National Curriculum and the Reception Year teaching programmes of the National Numeracy
However since the P Levels are intended as an assessment tool they describe performance
and don't provide detail of constituent skills. The reception teaching programmes tend
to assume children understand what counting is, and also assume that they have gathered
many fundamental skills such as pointing or appreciating sequences. Children with very
special needs may not have acquired these fundamentals.
In this article I propose to look closely at the "parts of counting".
When we are aware of the constituent skills of the counting process we will be able to
see how to:-
- Recognise where and how children are tripping up when we ask them to count.
- Emphasise the relevant parts for them in everyday activities.
- See how to prompt them constructively as we share learning experiences with them.
- Devise activities and games that help them practice the various parts of counting.
- Set realistic targets for their development.
The Parts of Counting
An obvious manifestation of counting happens as children learn to speak and use sounds that refer to quantity. Delighted adults begin to use rhymes stories and songs to teach them the number names, and are soon rewarded by children repeating them. Proud parents relate that they can count to three, later, ten and so on.
There is however much more to counting than repeating sounds. It entails doing many things at once like rubbing your tummy and patting your head, only more complicated and with a purpose you need to be aware of.
In the Language of Psychology
"When counting the child must co-ordinate the production of two continuous active sequences, saying the number words and producing points, while concurrently co-ordinating the points with a set of spatially distributed objects.
These requirements, accurate production of number words, plus their co-ordination in time with pointing and in space with objects allows considerable scope for error.
McEvoy J. (1989) "From Counting To Arithmetic". British Journal of Special Education.
Vol 16. No 3. Research Supplement, pp 107-110.
Knowing about the parts of counting, so that we can organise better learning opportunities for those who have difficulty with the spontaneous learning processes, will make an important contribution to their mathematical development.
For many children with severe learning difficulties the parts of counting will provide relevant content for the numeracy hour.
Knowing the number names
Though Piaget (1965) identified that children's recitation of number sequences and facts were mathematically meaningless later researchers suggest that such recitation is useful rehearsal. It establishes the sequence of words and prepares a framework that children relate to as they develop other counting skills.
The Initial Acquisition of number names
Though they may use number names referring to small quantities at an earlier age, children usually begin to acquire the sequence of number names around the age of two. (Fuson et al. 1982) 2. In their daily experience they hear and imitate the number sequence in rhymes song etc. and as adults deliberately count for them.
Evidence about how children acquire words from the context of their use suggests that initially they may not perceive the sequence as separate words, but as a relatively meaningless string of rhythmic sound.
Nevertheless this is a phase of acquisition, and their learning is assisted by the continuity and rhythm of the sound sequences, which cue and prompt sound recall, and helps to fix the order of sounds.
Elaborating on the list
Separating the sound flow into separate words, and establishing the order of those words comes as a later phase of elaboration. A further part of the elaboration phase is when children begin to use the words as number names and the words become "objects of thought" that symbolise quantity, and can be used for counting things. When this knowledge is absorbed the child will begin to recognise that there is a fixed order on which they can progress up and down, they will become able to say number names onwards or backwards from a given point.
To complicate matters different aspects of number sequences − teen structure 13 to 20 and the decade structure 20 to 90 may be being learned at the same time but be in different phases of acquisition. E.g. when one to ten has moved into the phase of elaboration, the teens may just be starting to be acquired as a sound sequence. The whole process takes a number of years and the rate at which children develop the skills is varied.
Learning the words of counting sequences
- Learning the sequence connected in a stream, of rhythmic sound.
- Beginning to separate the individual words, maintaining their order.
- Confirming the order of occurrence.
- Knowing the order backwards
- Knowing the sequence onward or backward, from a given point.
- Confirming the connection of individual words to a related quantity.
In order to count a group of objects children must be able to itemise them and
tag each with a number name. Schaffer et al (1974).
Most young children achieve this by pointing, and linking the word to the object as they point.
In order to do this accurately they must:
It is easy for them to make a mistake whilst:
- Pay attention to the objects.
- Control the motor activity - pointing.
- Make verbal output.
- Co-ordinate these actions in both space and time.
- As they do this they must remember which items have been counted and
which are left to count, this is called partitioning.
For young children touching is an important part of this process of itemisation, and provides physical prompt to help with timing the saying of the naming word. Martin Hughes (1986) notes how children still resort to pointing and tapping to assist their counting even when objects are out of sight. He also reminds us of the powerful use made of finger counting all over the world as a means of assisting both itemisation and partitioning.
- Controlling the physical act of pointing and controlling their attention to space.
- The timing of saying the number words at the right moment as they point.
This evidence should illustrate to us the importance of motor activities within counting strategies, and therefore the importance of activities that encourage and facilitate touching and pointing skills when we teach children who have difficulties.
Eventually children must move from physical partitioning and pointing and be able to partition mentally. There is evidence that as they do this they still use physical and rhythmic tactics such as finger counting and tapping.
Easy Counting Mistakes
- Fail to correspond their pointing to individual objects.
- Fail to correspond the sound with the pointing action.
- Miss an object.
- Itemise an object more than once.
- Missing a number name.
- Applying the same name twice.
- Confusing the order of names.
- Lose track of what has been counted and what remains to be counted.
- Don't stop the verbal sequence at the last object, keeping on because of the rhythm.
- Don't realise the last number is cardinal.
- Miss some objects because they don't think they should be included in the
count because of their colour, shape, position etc.
The Counting Principles
As they co ordinate the physical pointing and the verbal naming, to count accurately there are a number of rules that children need to apply. Five principles necessary for accurate counting were described by Gelman and Gallistel, (1978). The first three principles relate to "How to Count" and the last two are about "Applying Counting".
-- The One to One Principle --
Understanding and ensuring that each item receives one tag only
It is necessary to realise that the name tags are specially for counting with, they are nothing to do with other characteristics of the items being counted.
- Physically keeping track or mentally partitioning - which items have already been counted and those which remain.
- It also requires tagging - summoning up and applying distinct names one at a time.
In the early stages of learning to count children may be vague or imprecise about their pointing, they wave their fingers in the general direction, but let the rhythm of the verbal counting sequence dominate the speed at which they count, and consequently lose correspondence. Later they become more aware of the importance of co-ordinating the itemising and tagging, and they develop strategies for keeping track, and noticing if they have double counted or missed items. Gelman, R. Meck,E. (1983).
-- The Stable Order Principle --
The name tags must always be used in a stable order
This presents the child with the problem of remembering a long list. Bearing in mind the general limitations of short-term memory (Miller 1956), the human mind only being able to keep track on around seven items at once, we will recognise the valuable role of intonation and rhythm. They offer prompts, and connections that make memorable "chunks" and so help learning the number sequence.
-- The Cardinal Principle --
The final number represents the size of the set
When the child understands this principle they recognise that earlier numbers were temporary steps towards the last number tag, which is special, because it is the Cardinal Number and represents "how many" items have been counted. Appreciating the importance of Cardinality is an important milestone in a child's mathematical development, it is a keynote in understanding that the process of counting has a meaningful and useful purpose.
Fully grasping the cardinal principle depends on understanding the previous two principles, it therefore matures after them. There are three phases in its development
(Fuson & Hall 1983 ).
It is necessary for the child to grasp the cardinal principle before they will be able to:
- Reciting the last number with no clear idea that it relates to quantity, but because they realise it is the response the adult expects.
- Understanding that the last number of the count relates to the quantity.
- Understanding the progressive nature of cardinality i.e. if they are stopped in the middle of a count they can say how many they have counted so far, then carry on.
- Understand that the next number in the sequence represents a larger quantity.
- Use the technique of counting on.
- Use counting to determine and compare the equivalence of sets.
The next two principles are about what can be counted and applying counting.
-- The Abstraction Principle --
Counting can be applied to any collection − real or imagined
Adults realise that they may count physical or non- physical entities, objects that are not present, or even ideas. Young children on the other hand count physically present items and they group things in accordance with how they see their immediate relationships.
Variations in material properties or position may affect their view as to whether an item should be included in a count. What children of different ages or stages of development conceive as allowable within a counting sequence raises important considerations.
- What they might think about including or leaving out of the count on grounds of physical properties, position etc.
- Are they able to understand they can count objects they cannot see.
- Use counting to determine and compare the equivalence of sets.
- Can they count events as they happen, and events that occur elsewhere.
- Can they count ideas.
-- The Order Irrelevance Principle --
The order in which items are counted is irrelevant, the same cardinal value will be reached
This principle requires knowledge about the previous four principles grasping it entails understanding that:
It is necessary to grasp this principle of abstraction in order to be able to generalise the use of counting as a tool. It helps us confirm the consistency of the quantity of a set, and it is confidence in that consistency that enables us to be sure about making comparisons. Such confidence helps us to override the messages of perception that may confuse us when spatial changes make things appear bigger, and it may therefore underlie our ability to recognise the conservation of number.
- Each counted item is still a "thing" not a "one" or "two" etc.
- The name tags were temporarily given and do not necessarily adhere to the objects once the counting is finished.
- Whatever order the objects are counted in the same cardinal result occurs.
The Counting Principles:
How to Count
1. -- The One to One Principle --
2. -- The Stable Order Principle --
3. -- The Cardinal Principle --
4. -- The Abstraction Principle --
5. -- The Order Irrelevance Principle --
Some Useful Skills
Children and adults do not always count in order to make out the size of a group, when the group is small they can do this by a process of pattern recognition. Most people will
recognise that this is what happens when a dice is rolled, the process is known as Subitising.
This early ability enables children to make out and compare small groups and may support the refinement of concepts of quantity. It only works with small groups and this may account for why quite young children can be accurate with processes involving small numbers, but lose track when the groups involved are larger than the perceptual range of subitising.
Early researchers believed it to be a low level activity but Gelman proposes that it uses complex pattern recognition skills. She suggests it is a strategy that reduces load on short term memory during such processes as making comparisons, and combining groups, and continues to be used in adult life.
So although at first it might primarily be useful for recognising the size of small groups it may also become a useful tool for visualising, comparing, working out, and estimating. Gelman suggests that the ability to subitise is refined through the processes of learning to count.
Children with severe learning difficulties often have difficulty with the translation of symbolic representations such as learning to read. Paradoxically symbol systems like Makaton and Rebus are often helpful offering prompts that help them separate and make sense of the information flow within communications.
Symbolic representation becomes a vital part of the language of mathematics, and as with reading working with numerals often presents difficulties for SLD pupils. Whilst a discussion of symbolic development would take more space than is available a brief overview of strategies children use to keep track during counting, and aid the making of comparisons, may illustrate how some forms of symbolism may help our special pupils.
Perhaps the first symbols which children use are their fingers, they keep tally with them during a count using them as symbols to represent objects. Fingers are very useful in that they can be both objects and symbols (Hughes 1986). They can be used to keep tally but are also useful to provide a visual aid representing the parts of simple arithmetic processes. Using fingers may be seen as a development beyond itemising objects. Their use extends the pointing and touching we discussed earlier, and as such may be a link between sensorimotor learning and symbolic representation.
Children who are unable to use their fingers are disadvantaged in experiencing this, and it may be helpful to them if teachers use touch as a communication medium as they count with these children.
It is self evident that making marks to represent things is important because it is a means of recording and remembering. What may be less evident to us an yet vitally important is the role that mark making plays in helping children move towards abstract thinking, drawings become symbols, and children come to understand that representations and symbols can "stand for things" we can even represent things that are not here right at this moment!
Children's progress toward written representation of number passes through three stages, the first two relate strongly to the -- One to One Principle --.
Where objects being counted are drawn depicting some of their characteristics as well as the numerosity, three bricks may look like this:
Where simple marks are used to tally or represent the objects. Despite being quite abstract iconic representations are used even by pre school children, perhaps because it relates to using fingers and is useful for counting events:
At first idiosyncratic, then conventional numerals and later words:
The relationship between pictorial representation and making a mental visualisation of a quantity can easily be understood, and how it might help as a memory strategy, or provide visible evidence for the child to work on will also be clear.
The use that children make of their fingers is apparent to any observer and the development of keeping tally by making iconic marks can be seen as a step extending finger counting. Both provide a mode of symbolisation which helps the child retain focus firstly on accounting for if an object is there or not, and secondly as a cumulative record which helps memory and can be worked on later.
Pictorial and Iconic recording help children keep a track on quantities and provide concrete visual records for them. It may be useful to encourage children to use such systems of representing quantities and keeping tally to support counting activities. It seems likely that they need to be fully grasped before the relationships between quantity and more symbolic numerals can be realised.
Translating the Problem
Looking back over the skills and processes I have described it is easy to regard the course of learning as a linear sequence where skills develop in a specific order and we should teach one skill at once. However such a view is too simple, children have surprising abilities at ages much earlier than a rigid view of the sequence would lead us to expect for example:
These examples might be taken to indicate that children have both innate understandings of quantity, and a disposition towards using mathematical thinking, even before they can count and express numerical comparisons. Exposure to experiences enables them to use their early concepts and abilities in parallel, each contributing their part to the refinement and consolidation of the way that they are used together.
- There is evidence that children have appreciation of quantities at very early ages. They are sensitive that three objects are more than two as early as four months. (Starkey et al 1983) i.e. long before they count or even itemise.
- Infants have an intuitive perception that addition increases and taking away decreases, they naturally use their body parts for keeping tally, and supporting their own systems of counting small groups (Geary 1994).
- Vygotsky (1978) described children's spontaneous use of mathematical ideas and processes as they related directly to their own lives, e.g. using numbers which had personal relevance, or inventing their own ways of writing numbers
This process is reflected in the development of counting. For example children who have not yet mastered number names and their order still practice itemising and enumerating, using their fingers and their own versions of number names. As they develop their knowledge of number names their perceptions of quantity help them to understand order and comparative value etc. Many people assume that children do not learn to add or subtract until they can count, but counting itself is a process of addition, and counting backwards is subtraction.
Children can apply mathematical concepts to practical tasks long before they can express their understanding. Martin Hughes (1986) describes how pre school children were able to add and subtract small groups when the words used in the question related to real items. In contrast they were unable to answer when the reference to the objects was left out. He suggests that relating the problem to real objects allowed the children to visualise the items and use their practical knowledge to give the answer. Without reference to real objects the child was puzzled by the question, and could not translate the mathematical language into real terms. E.g.
Question: "What is two and one more?"
Child's response: "One more what?"
He noted that even basic mathematical language could be confusing because familiar words had different implications when used to relate to number.
Question: "What is the difference between 6 and 11?"
Child's answer: "Six is curly"
He noted that children needed to be able to "realise when the language of mathematics
was being spoken."
The insights that Hughes work offers us are:
- Children's concepts of mathematical operations have a practical basis.
- Many difficulties children have with mathematical problems relate to translating between their concrete understanding and the language that is used for mathematical purposes.
Approaching the Teaching of Counting
Throughout the article we have discussed the complexity of what is usually regarded as a simple operation. We have seen that the parts of counting, - includes physical co ordination, mental and verbal skills, and learning rules. We have also seen that in order to understanding mathematical language children need to relate it to real objects, or mental images of them. These aspects of learning have practical and presentational implications that we need to bear in mind when we arrange how to present mathematical experiences to help our children learn, including as we teach them to count.
The framework of our teaching needs to be reality, we need to exploit everyday activities, create events where counting takes place, and games that give opportunities to notice, practise and integrate the constituent skills. The analysis of constituent skills made in this article can provide a framework through which we can add detail that will enhance the National Curriculum programs of study. The knowledge that has been outlined about children's inclination towards using their concrete experience, and how they use visualisation, should provide guidance to us. It illustrates that we need a style of delivery that refers to real events and illustrates the language that is used when referring to changes in quantity.
- How well they can carry out the physical and verbal operations of counting.
- How well do they grasp the "principles of counting". We need to remember the extent to which children rely upon real objects to visualise changes in quantity.
- Are they able to make pictorial or symbolic representations of things that they count?
- Finally we need to help them learn to translate between their practical knowledge and language that is used to refer to mathematics.
The information that has been highlighted in the article will help teachers see targets that are aspects within the skill of counting. Armed with this level of consciousness about detail of counting processes the teacher will be able to maximise many opportunities within everyday situations. Develop events or games when there is opportunity to tally; record; match; visualise; compare; name etc.
Knowledge of detail will enable them to set the child realistic challenges, or offer support to ensure that the child learns from the process, rather than fails by not being able to co ordinate all the parts to produce the right answer.
- Piaget, J. (1965) The Childs Conception of Number. NY.WW Norton.
- Fuson,K.C., Richards, J.and Moser, J.M. (1982) The acquisition and elaboration of the number word sequence. In Brainerd, C. Progress in cognitive development: Children's logical and mathematical cognition, Vol.1. NY:Springer Verlag.
- Schaffer, B. Eggleston, V.H.and Scott, J.C. (1974) Number Development in young children Cognitive Psychology 6 357-379
- Gelman,R. Gallistel, R.C. (1978) The Childs Understanding of Number. Cambridge. USA. Harvard.U.P.
- Gelman, R. Meck,E. (1983) Prescchoolers Counting: Principles before skills. Cognition, 13, 343-359.
- Miller, G.A. (1956) The Magical number 7 + or − 2 Psychological Review, 63, 81-97.
- (Fuson. KC & Hall JW ( 1983) The acquisition of early number word meanings. A conceptual analysis and review. In Ginsburg, H.P. (ed) The development of mathematical thinking. NY Academic Press.
- Starkey,P. Spelke, E.S. Gelman, R. (1983) Detection of Intermodal Numerical Correspondences by Human Infants. Science 222, pp 179 -181
- Geary,D.C. (1994) Children's Mathematical Development: Reearch and practical applications. Washington DC. American Psychological Association.
- Vygotsky, L.S. (1978) Mind in Society: The Development of Higher Psychological Processes. Translated by Cole, M. Cambridge,Mass. Harvard U.P.
- Hughes, M. (1986) Children and Number. Pgs 24-36. Oxford. Blackwell.
- Guidelines for Planning teaching and assessing the curriculum for pupils with learning difficulties were published by QCA in 2001. Read more about these Guidelines here.
Retired as the head teacher of Turnshaws Special School in Kirklees following
an outstanding Ofsted report.
He has thirty years teaching experience in mainstream and special education.
He now works as a freelance trainer and consultant.